Nonlinear dynamic phenomena

Finally, Section 2.4 analyses a simplified model of a bursting pancreatic /3-cell [12]. The purpose of this section is to underline the importance of complex nonlinear dynamic phenomena in biomedical systems. Living systems operate under far-from-equilibrium conditions. This implies that, contrary to the conventional assumption of homeostasis, many regulatory mechanisms are actually unstable and produce self-sustained oscillatory dynamics. The electrophysiological processes of the pancreatic /3-cell display (at least) two interacting oscillatory processes A fast process associated with the K+ dynamics and a much slower process associated with the Ca2+ dynamics. Together these two processes can explain the characteristic bursting dynamics in the membrane potential. 

As illustrated in Section 2.4, the significance of nonlinear dynamic phenomena seems to be even more pronounced at the cellular level. The insulin-producing... 

Thermal frontal polymerization exhibits the full range of nonlinear dynamics phenomena, including those driven by hydrodynamics as well as driven by intrinsic feedbacks in the chemistry. Features unique to polymerization kinetics and properties allow the study of convection in fronts and novel spin modes. 

Are there new nonlinear dynamical phenomena that arise because of the special properties of polymer systems ... 

Attractive in its simplicity, yet complex in its behavior, the Continuous Stirred Tank Reactor has, for the better part of a century, presented the research community with a rich paradigm for nonlinear dynamics and complexity. The root of complex behavior in this system stems from the combination of its open system feature of maintaining a state far from equilibrium and the nonlinear non-monotonic feedback of various variables on the rate of reaction. Its behavior has been studied under various designs, chemistries and configurations and has exhibited almost every known nonlinear dynamics phenomenon. The polymerization chemistry has especially proven fruitful as concerns complex dynamics in a CSTR, as attested to by the numerous studies reviewed in this chapter. All indications are that this simple paradigm will continue to surprise us with many more complex discoveries to come. 

For modelling conformational transitions and nonlinear dynamics of NA a phenomenological approach is often used. This allows one not just to describe a phenomenon but also to understand the relationships between the basic physical properties of the system. There is a general algorithm for modelling in the frame of the phenomenological approach determine the dominant motions of the system in the time interval of the process treated and theti write... 

What will happen further away from equilibrium when the linear relationship between cause and effect breaks down is clear from the above example but for other instances is the subject of much research and speculation. With nonlinear relationships, the scope of phenomena becomes nearly unpredictable. Nonlinear dynamics [9,10] may well provide the clue to the phenomenon of macroscopic complexity [11], a rapidly expanding field of science, defined by some [12] as quickly becoming a field of "perplexity."... 

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

One of the most promising applications of nonlinear dynamics to polymer science is the phenomenon of frontal polymerization (Section III). Frontal pol)mierization is a process of converting monomer into polymer via a localized reaction zone that propagates through the monomer. There are two modes of frontal polymerization. 

The mathematical objects of nonlinear dynamics are models — explicitly defined dynamical systems depending on a finite number of parameters. The primary quality of a model is that it must properly, at least qualitatively, describe the nature of the associated physical phenomenon. The primary goal in the study of a model is to give a rigorous mathematical explanation. In this connection let us recall the following remark made by Lyapunov ... it is not permitted to use dubious arguments as soon we have started to solve a specific problem from mechanics or physics, it does not matter, whatever is set up correctly from the point-of-view of mathematics. As soon as the system is defined it becomes a problem of pure analysis and must be treated as such. ... 

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by a nonlinear (inhomogeneous) external periodic excitation (as regards the coordinates of the excited system) and is remarkable for the occurrence of the following objective regularities the phenomenon of discrete oscillation excitation in macro-dynamical systems having multiple branch attractors and strong self-adaptive stability. 

The main goal of this report is to present a phenomenon of highly general nature manifested in various dynamical systems. We present the occurrence of peculiar quantization by the parameter of intensity of the excited oscillations and we show that given unchanging conditions it is possible to excite oscillations with a strictly defined discrete set of amplitudes the rest of the amplitudes being forbidden . The realization of oscillations with a specific amplitude from the permitted discrete set of amplitudes is determined by the initial conditions. The occurrence of this unusual property is predetermined by the new general initial conditions, i.e. the nonlinear action of the external excited force with respect to the coordinate of the system subjected to excitation. 

The chaotic behavior is an interesting nonlinear phenomenon which has been intensively studied during last two decades. The deterministic techniques have been used to understand the d3mamical structure in several nonlinear systems [5], [6], [42]. Particularly, the two-phase flow systems present nonlinear d mamical behavior which can be studied by means of chaos criteria behavior [2], [17], [25], [31], [45], [51]. Two-phase flows they provide a rich variety of cases whose dynamics lead to oscillatory patterns. The following published results are example ... 

This phenomenon of increased conversion, yield and productivity through deliberate unsteady-state operation of a fermentor has been known for some time. Deliberate unsteady-state operation is associated with nonautonomous or externally forced systems. The unsteady-state operation of the system (periodic operation) is an intrinsic characteristic of this system in certain regions of the parameters. Moreover, this system shows not only periodic attractors but also chaotic attractors. This static and dynamic bifurcation and chaotic behavior is due to the nonlinear coupling of the system which causes all of these phenomena. And this in turn gives us the ability to achieve higher conversion, yield and productivity rates. 

In Section IV.B.4 we have shown that the quadratic dynamic susceptibilities of a superparamagnetic system display temperature maxima that are sharper than those of the linear ones. If the maximum occurs as well at the temperature dependence of the signal-to-noise ratio, this should be called the nonlinear stochastic resonance. However, before discussing this phenomenon, one has to define what should be taken as the signal-to-noise ratio in a nonlinear case. 

An analysis of the recent observation data [30,31] shows that baroclinic Rossby waves that are generated off the eastern coasts in the northern parts of the Pacific and Atlantic oceans in a period of about a year represent their dominant non-stationary dynamical response to the annual cycle of the atmospheric forcing in the latitudinal range from 10-15° to 45-50°N. In so doing, their mean phase velocities (0.02-0.03 ms 1 at 40-45°N) are higher than the theoretical values (about 0.01 ms-1). A similar situation is observed in the Black Sea as well [27]. In [32], several reasons of this phenomenon were listed such as the interaction with more large-scale non-stationary processes, topographic and nonlinear effects, and insufficient duration and spatiotemporal resolution of the observation data. 

The phenomenon of self organization occurs at nonstabHities of the sta tionary state and leads to the formation of temporal and spatio temporal dissipative structures. Remember that oscillating instabilities of stationary states of dynamic systems can be observed for the intermediate nonlinear stepwise reactions only, when no fewer than two intermediates are involved (see Section 3.5) and at least one of the elementary steps is kinet icaUy irreversible. The minimal sufficient requirements for the scheme of a process with temporal instabilities are not yet strictly formulated. However, in aU known examples of such reactions, the rate of the kineti caUy irreversible elementary reaction at one of the intermediate steps is at least in a quadratic dependence on the intermediate concentrations. Among these reactions are autocatalytic steps. 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

Synchronization is a fundamental phenomenon found in nonlinear oscillatory systems [12]. The most prominent example, known since long time ago (Huygens, 1665), is the adjustment to a common frequency of two pendulum clocks with slightly different frequencies, coupled via a common support. This type of synchronization between two coupled systems is called mutual synchronization. In the following we are interested in the synchronization of a system to a periodic driving, called forced synchronization. In a synchronized state the systems dynamics is entrained to the signal, i.e. the system inherits the very same frequency of the signal (1 1 synchronization) or the frequencies are locked with some rational n m relation. This... 

For systems of more than two degrees of freedom, however, their role is not so obvious because their dimension is not large enough to work as barriers. This problem is closely related to the phenomenon of Arnold diffusion. In other words, dynamical processes along nonlinear resonances may create a way to go around these tori. In particular, intersections of resonances would play a dominant role in IVR since chaotic diffusion... 

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